What about to use replaceAll java.lang.String method:

    String str = "qwerty-1qwerty-2 455 f0gfg 4";      
    str = str.replaceAll("[^-?0-9]+", " "); 
    System.out.println(Arrays.asList(str.trim().split(" ")));

Output:

[-1, -2, 455, 0, 4]

Description

[^-?0-9]+

• [ and ] delimites a set of characters to be single matched, i.e., only one time in any order
• ^ Special identifier used in the beginning of the set, used to indicate to match all characters not present in the delimited set, instead of all characters present in the set.
• + Between one and unlimited times, as many times as possible, giving back as needed
• -? One of the characters “-” and “?”
• 0-9 A character in the range between “0” and “9”

Instead of replacing characters with empty string, replace with a space. And then split over it.

import java.util.ArrayList;
import java.util.List;

public class Main {
    public static void main(String[] args) {
        String line = "First number 10, Second number 25, Third number 123 ";
        String numbersLine = line.replaceAll("[^0-9]+", " ");

        String[] strArray = numbersLine.split(" ");

        List<Integer> intArrayList = new ArrayList<>();

        for (String string : strArray) {
            if (!string.equals("")) {
                System.out.println(string);
                intArrayList.add(Integer.parseInt(string));
            }
        }

        // what I want to get:
        // int[0] array = 10;
        // int[1] array = 25;
        // int[2] array = 123;
    }
}

This would be a suitable solution too:

String[] input = JOptionPane.showInputDialog("What are your grades of this month?").split(" ");
        double[] grades = new double[input.length];
        double average = 0.0;
        for (int i = 0; i < input.length; i++) {
            // Note that this is assuming valid input
            grades[i] = Double.parseDouble(input[i]);
            average+=grades[i];
        }
        average /= grades.length;

So you could type in multiple "grades" seperated by a whitespace.

if the numbers are allways separated by x^, just split the string using this separator

String[] splitted = "3x^2".split("x\\^");

Just split your input according to one or more non-digit characters and then convert the datatype of each element to integer.

String[] parts = str.split("\\D+");

What @jmoreno has said is pretty much spot on but seeing as this is a java problem remember that the convention is to use lower case letters for method names.

The important things to remember are that each method should be simple, readable, encapsulate a unit of work and be easily testable. Again, following what @jmoreno has said and considering the gatherInput method, you can take the first five lines of code and create a new method that returns your string input. If you are using Eclipse this is very easy, highlight the code and click Alt+Shift+M which will open the 'Extract Method' refactor dialog box.

The loop is the only real place where you have to think a little more as currently it is doing multiple jobs, but if you break out the code that extracts the numbers (extractIntegers(input)) then the rest becomes easy - you should also be able to remove the additional index == input.length() test.

Other considerations:

1. Have you thought about a for loop instead of a while loop? All of your cases increment the index by 1.
2. Do you know how the input string should be formatted, in which case you could use a regular expression to extract the numbers.
3. You have used isDigit once, why have you mixed it with the >= <= solution? What about the letters?

You can use String.split():

String[] nbs = str.split("[^0-9]+");

This will split the String on any group of non-numbers digits.

Using the code as you currently have:

Object[][] database = new Object[14][5];

We can add a few things to make it more friendly to the reader

final int STUDENTS = 14;
final int GRADES = 4;
Object[][] database = new Object[STUDENTS][GRADES + 1];

In order to be able to check the average values

public double getStudentAverageGrade(Object[] student)
{
    for(int i = 0; i < GRADES; i++) // forEach grade
    {
        gradesSummatory += (double) student[i + 1]; // plus one is added since we ignore first value (which is students name)
    }
    return gradesSummatory/GRADES;
}

Now, to be able to see the average of ALL the students, its just easier since we simply have to implement the method previously created.

public double getAverageGrade()
{
    double avgSummatory = 0;
    for (int i = 0; i < STUDENTS; i++)
    {
        avgSummatory += getStudentAverageGrade(database[i]);
    }
    return avgSummatory/STUDENTS;
}

By saying this I don't consider it being a really optimal solution (you could sum all the elements and divide only once) but it's the most expressive non-object oriented solution imo. I'd rather abstract these elements/objects into different classes such as Student so as to reuse code as much as possible. But for this case, this could do the trick.